Potential theory for the sum of the 1-Laplacian and p-Laplacian

The variational problem of minimizing the functional u↦∫Ω|Du|+1p∫Ω|Du|p−∫Ωau on a domain Ω⊂Rn under zero boundary values, which appears for example in the theory of Bingham fluids, shows interesting phenomena such as the formation of a subset of Ω with positive measure where the minimizer is constant (a “plateau”). The corresponding Euler-Lagrange equation should be −Δ1u−Δpu=a, but the 1-Laplacian is not defined in points where Du=0. We show that it is nevertheless possible to define subsolutions and supersolutions for this equation. We show that the comparison principle is valid, and we begin developing a potential theory analogous to what is known about the p-Laplacian. In addition we apply those notions to rotationally invariant solutions, and we discuss properties of the plateau and a possible connection to the theory.